# On the Boundary between Decidability and Undecidability of Asynchronous Session Subtyping

## TL;DR

This paper explores the boundary between decidability and undecidability in asynchronous session subtyping, identifying a more applicable decidable fragment without buffer limitations or restrictions on multiple choices.

## Contribution

It introduces the first decidable fragment of asynchronous session subtyping that allows unlimited buffers and multiple choices, expanding practical applicability.

## Key findings

- Decidability achieved without buffer size restrictions.
- Undecidability persists even with covariance and contravariance restrictions.
- Identifies the boundary between decidable and undecidable subtyping fragments.

## Abstract

Session types are behavioural types for guaranteeing that concurrent programs are free from basic communication errors. Recent work has shown that asynchronous session subtyping is undecidable. However, since session types have become popular in mainstream programming languages in which asynchronous communication is the norm rather than the exception, it is crucial to detect significant decidable subtyping relations. Previous work considered extremely restrictive fragments in which limitations were imposed to the size of communication buffer (at most 1) or to the possibility to express multiple choices (disallowing them completely in one of the compared types). In this work, for the first time, we show decidability of a fragment that does not impose any limitation on communication buffers and allows both the compared types to include multiple choices for either input or output, thus yielding a fragment which is more significant from an applicability viewpoint. In general, we study the boundary between decidability and undecidability by considering several fragments of subtyping. Notably, we show that subtyping remains undecidable even if restricted to not using output covariance and input contravariance.